Measures of Competion in banking industry

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1 
 
 
Measures of competition and concentration in the banking 
industry: a review of the literature 1 
 
 
 
 
J.A. Bikker and K. Haaf 2 
 
 
 
 
Research Series Supervision no. 27 
 
De Nederlandsche Bank 
 
 
 
September 2000 
 
Abstract 
Measures of concentration and competition are of vital importance for welfare-related public policy 
toward market structure and conduct in the banking industry. Theoretical characteristics of ten market 
concentration measures are discussed and numerical examples illustrate differences and similarities 
between these concentration indices in operation. Measures of competition can be divided into 
structural and non-structural ones. In structural approaches concentration ratios take a central 
position in order to describe the market structure, forging a natural link between concentration and 
competition. The impact of market concentration on market performance has its roots in both the 
oligopoly theory and the structure-conduct-performance paradigm. Non-structural approaches to 
measuring competition do not depend on concentration. The survey includes empirical results of the 
various methods applied to the European banking industry. 
 
 
1 Appeared as: J.A. Bikker, K. Haaf, 2002, Measures of competition and concentration in the banking 
industry: a review of the literature, Economic & Financial Modelling 9, 53-98. Reprinted in: J.A. Bikker, 2004, 
Competition and efficiency in a unified European banking market , Edward Elgar, 315 pages. 
2 Section Banking and Supervisory Strategies, Directorate Supervision, De Nederlandsche Bank. 
 2 
 
1 INTRODUCTION 
 
The deregulation of financial services in the European Union, the establishment of the Economic and 
Monetary Union (EMU) and the development of information technology are expected to contribute to 
dramatic changes in European banking markets over the coming years, with vast implications for 
competition and concentration in the banking and financial sector. The recent wave of mergers in the 
European banking industry can be seen as a result of these developments. This process of 
consolidation may affect competition, particularly on local retail markets, and enhances concentration, 
whereas the size of new global players may cause concerns about financial stability. 
 
To assess the implications of these developments, it is necessary to investigate the impact of 
consolidation on the market structure and the performance of banks. Measures of concentration and 
competition are essential for welfare-related public policy toward the banking market. In recent years, 
however, only a limited number of empirical studies have investigated competition and concentration 
in European banking markets. The scarcity of analysis in this field is primarily due to a dire shortage 
of detailed European sub-market banking data. This study provides an overview of the tools at hand to 
investigate competitive conditions and to calculate the degree of concentration in (banking) markets, 
as presented in the literature. It summarises the theoretical foundations of the various measures of 
competition and concentration as well as their empirical application to the European banking industry. 
 
Concentration and competition are linked to product markets and geographical areas, both in theory 
and in empirical analyses. Banks provide a multitude of products that do not serve a unique market, 
and defining a relevant market involves making a preliminary decision about potentially relevant 
structural characteristics, such as concentration and competition (Kottmann, 1974). The relevant 
market includes all suppliers of a good who are actual or potential competitors, and it has a product 
dimension and a geographical dimension. The product definition of a market requires the 
determination of the range of products which can be assigned to a particular market on the basis of 
their substitutability in terms of consumer demand. Likewise, the geographical boundaries of a market 
are drawn according to existing and potential contacts between actual and potential market 
participants. They are determined from the customer’s point of view and take into consideration 
individual consumer as well as product characteristics. The mobility of banking customers, and 
therefore the geographic boundaries of the market, depend quite crucially on the type of customer and 
their economic size; the local dimension of a market is relevant for retail banking products and the 
regional or international dimension is relevant for corporate banking. Product characteristics influence 
the mobility of customers in that commercial borrowers tend to display greater mobility in their search 
for financing possibilities than depositors (Kottmann, 1974, Deppe, 1978, and Büschgen, 1993). 
 
 3 
 
The chapter is organised as follows. Section 3.2 presents an overview of market concentration 
measures, discusses their theoretical characteristics and gives numerical examples to illustrate 
differences and similarities between concentration indices in operation. The literature on the 
measurement of competition can be divided into two major streams: structural and non-structural 
approaches. In the structural approaches, concentration ratios take a central position in order to 
describe the market structure. Structural approaches are investigated first (in Section 3.3), as they 
constitute a natural link between concentration and competition. The impact of market concentration 
on market performance has its roots in both the oligopoly theory and the structure-conduct-
performance (SCP) paradigm. Whereas the former approach is founded in economic theory, the latter 
is an ad hoc approach, often criticised in methodological as well as identificational terms, but 
nevertheless applied empirically ad infinitum. Where a SCP model can be based on any concentration 
ratio, two formal derivations of the competition-concentration relationship based on oligopoly theory 
each exactly define the relevant concentration ratio. This section also surveys empirical results of the 
SCP test for the European banking industry. Section 3.4 provides a theoretical presentation of non-
structural approaches to the measurement of competition, some of which share the same theoretical 
roots as the formal structural methods of the preceding section. Furthermore, this section summarises 
the results of the various studies applying those approaches to the European banking industry. 
Appendix 3 reviews the most important distributional concepts employed in this chapter. 
 
2 MEASURES OF CONCENTRATION 
 
The importance of concentration ratios arises from their ability to capture structural features of a 
market. Concentration ratios are therefore often used in structural models explaining competitive 
performance in the banking industry as the result of market structure.3 This feature will be discussed in 
Section 3.3, where the impact of concentration on competitive performance is investigated. 
Concentration ratios are also able to reflect changes in concentration as a result of the entry of a bank 
into the market or its exit from it, or caused by a merge. This feature is used in the US, for instance, in 
the enforcement process of anti-trust laws in banking. 
 
The concept of industrial concentration has been extensively treated and lively debated in the 
economic literature. Despite the many different approaches to its measurement, general agreement 
prevails about the constituting elements of concentration measures, i.e. the number of banks (fewness) 
and the distribution of bank sizes (inequality)in a given market. However, the cla ssification of 
 
3 It should be noted, however, that a measure of concentration does not warrant conclusions about the 
competitive performance in a particular market. Even in a highly concentrated market, competitive behaviour 
between the leading banks is still possible. 
 4 
 
concentration measures in the literature is not systematic. This chapter presents concentration indices 
(CI) exhibiting the general form: 
 
(2.1) å ==
n
i ii
wsCI
1
 
 
where is is the market share of bank i, iw is the weight attached to the market share and n is the 
number of banks in the market in question. This section considers ten concentration ratios – the k bank 
Concentration Ratio ( kCR ); the Herfindahl-Hirschman Index (HHI); the Hall-Tideman Index (HTI); 
the Rosenbluth Index (RI); the Comprehensive Industrial Concentration Index (CCI); the Hannah and 
Kay Index (HKI); the U Index (U); the multiplicative Hause Index (Hm); the additive Hause Index 
(Ha); and the Entropy measure (E) – and discusses the weighting scheme and structure of each. The 
theoretical discussion of the properties of the various indices is illus trated by their empirical 
application to the Dutch mortgage market. In addition, the most frequently used indices, kCR and 
HHI, are applied to 20 countries in order to assess their ability to measure and, more particularly, to 
rank bank concentration in these countries. 
 
2.1 Classification by weighting scheme and structure 
 
Concentration measures can be classified according to their weighting schemes and structure. Marfels 
(1971a) and Dickson (1981) discuss the weighting schemes of a number of concentration ratios. The 
weighting scheme of an index determines its sensitivity towards changes at the tail-end of the bank 
size distribution. Marfels differentiates between four groups of weights:4 
 
1. Weights of unity are attached to the shares of an arbitrarily determined number of banks ranked in 
descending order ( 1=iw , ki £" ), and zero weights are attached to the remaining banks in the 
industry ( 0=iw , ki >" ). An example is the k bank concentration ratio, probably the most 
frequently used concentration ratio. 
2. Banks’ market shares are used as their own weights ( ii sw = , i" ), so that greater weights are 
attached to larger banks. These indices take account of all banks in the industry. An example is the 
Herfindahl-Hirschman index, well-known from both theory and practice. 
3. The rankings of the individual banks are used as weights ( iwi = , i" ), where banks can be ranked 
in ascending or descending order. All banks are included in computing this index. Examples are 
the Rosenbluth index and the Hall-Tideman index. 
 
4 Instead of Marfels’ fourth group of weighting schemes, in which the shares of individual banks are used as 
weights in a weighted geometric mean, the present chapter presents the logarithm of the market share of the 
 5 
 
4. Each market share is weighted by the negative of its logarithm ( ii sw log-= , i" ). A smaller 
absolute weight is thus attached to larger market shares. An example is the Entropy index. 
 
Dickson’s (1981) approach to weighting schemes is somewhat more theoretical. By defining 
conjectural variation elasticity,5 he presents the derivation of various weighting schemes embedded in 
microeconomic theory. Dickson found only six of the 15 indices under investigation consistent with 
theoretical market models: the kCR , the HHI, the CCI, the mH , the HKI and the U. Some of the 
indices presented in this section are more complex and hence consistent with more than one of the 
weighting schemes; an example is the CCI. 
 
The structure of concentration indices can be discrete or cumulative. Discrete measures of 
concentration correspond to the height of the concentration curve at an arbitrary point.6 The k bank 
concentration ratio, for instance, belongs to this class of discrete measures. Practical advantages of 
discrete measures are simplicity and limitation of required data. In the literature, both supporters and 
critics of discrete concentration measures are numerous. Both parties, however, stress the impact of 
concentration on the market behaviour of banks. Supporters of discrete measures maintain the view 
that the behaviour of a market dominated by a small number of banks is very unlikely to be influenced 
by the total number of enterprises in the market: the calculation of concentration indices on the basis 
of the entire bank size distribution would be unnecessarily large-scale, while only marginally changing 
the final results. Critics adhere to the view that every bank in the market influences market behaviour 
and stress a severe disadvantage of discrete indexes: they ignore the structural changes in those parts 
of the industry which are not encompassed by the index of concentration. The competitive behaviour 
of the smaller market players might force the larger players to act competitively as well. 
 
Cumulative or summary measures of concentration, on the other hand, explain the entire size 
distribution of banks, implying that structural changes in all parts of the distribution influence the 
value of the concentration index. Of the measures presented in this section, the HHI, the CCI, the RI 
and the HTI as well as the E display this feature. Marfels (1971b) showed that it is possible to find 
corresponding measures of inequality for every summary measure of concentration, but that discrete 
measures of concentration do not exhibit this duality. Hence, he argues that the behaviour of discrete 
measures resulting from changes in the number of banks in the industry or in bank size disparity 
cannot be clearly identified. 
 
 
respective bank (relevant for the calculation of e.g. the Entropy measure). The exponential of the Entropy 
measure, which uses the former weighting concept, will not be discussed in this chapter. 
5 The concept of conjectural variation is explained in Section 3.2 below equation (3.5). 
6 Appendix 3 explains the concentration curve. 
 6 
 
2.2 Concentration Ratios 
 
2.2.1 The k bank concentration ratio 
Simplicity and limited data requirements make the k bank concentration ratio one of the most 
frequently used measures of concentration in the empirical literature. Summing only over the market 
shares of the k largest banks in the market, it takes the form: 
 
(2.2) å ==
k
i ik
sCR
1
 
 
giving equal emphasis to the k leading banks, but neglecting the many small banks in the market. 
There is no rule for the determination of the value of k , so that the number of banks included in the 
concentration index is a rather arbitrary decision. The concentration ratio may be considered as one 
point on the concentration curve, and it is a one-dimensional measure ranging between zero and unity. 
The index approaches zero for an infinite number of equally sized banks (given that the k chosen for 
the calculation of the concentration ratio is comparatively small as compared to the total number of 
banks) and it equals unity if the banks included in the calculation of the concentration ratio make up 
the entire industry. If the industry consists of n equally sized banks, then == å =
k
i ik
sCR
1
 
nkn
k
i
=å =11 , which is a decreasing function of the number of banks in the market, and which 
yields the numbersequivalent as ke CRkn = (White, 1982).
7 
 
2.2.2 The Herfindahl-Hirschman Index 
The Herfindahl-Hirschman Index (HHI) is the most widely treated summary measure of concentration 
in the theoretical literature and often serves as a benchmark for the evaluation of other concentration 
indices. In the United States, the HHI plays a significant role in the enforcement process of antitrust 
laws in banking. An application for the merger of two banks will be approved without further 
investiga tion if the basic guidelines for the evaluation of the concentration in deposit markets are 
satisfied. Those guidelines imply that the post-merger market HHI does not exceed 0.18, and that the 
increase of the index from the pre-merger situation is less than 0.02 (Cetorelli, 1999). Often called the 
full-information index because it captures features of the entire distribution of bank sizes, it takes the 
form: 
 
(2.3) å ==
n
i i
sHHI
1
2 
 
 
7 See Appendix 3 for an explanation of the numbers equivalent. 
 7 
 
which is the sum of the squares of bank sizes measured as market shares. As mentioned above, the 
Herfindahl-Hirschman index stresses the importance of larger banks by assigning them a greater 
weight than smaller banks, and it incorporates each bank individually, so that arbitrary cut-offs and 
insensitivity to the share distribution are avoided. The HHI index ranges between n1 and 1, reaching 
its lowest value, the reciprocal of the number of banks, when all banks in a market are of equal size, 
and reaching unity in the case of monopoly. Davies (1979) analyses the sensitivity of the HHI to its 
two constituent parts, i.e. the number of banks in the market and the inequality in market shares among 
the different banks and finds that the index becomes less sensitive to changes in the number of banks 
the larger the number of banks in the industry. 
 
Several authors propose linking the Herfindahl-Hirschman index to distributional theory by presenting 
it in terms of the moments of the underlying bank size distribution. A first attempt to present the HHI 
in terms of the distribution’s mean and variance has been undertaken by Adelman (1969). Kwoka 
(1985) rewrites the Herfindahl-Hirschman index as å = -+=
n
i i
sssHHI
1
2)( by defining the mean 
market share as ns 1= . Re-arranging yields the HHI as an inverse function of the number of banks in 
the industry and a direct function of the market shares’s variance about the mean: 
 
(2.4) 2)/1( snnHHI += 
 
This presentation points up two features of the Herfindahl-Hirschman index. First, the relation 
between the number of banks and the value of HHI is not a simple one. Given the number of banks in 
the market, the HHI increases with the variance, which is itself a function of the number of banks in 
the market (Adelman, 1969). Second, this presentation of the HHI is ambiguous in that a variety of 
combinations of the number and sizes of banks can produce the same HHI (Kwoka, 1985). However, 
this need not be a shortcoming and, for instance, holds also for the Lerner index.8 
 
In the context of hypothetical market analysis, Rhoades (1995) argues that the inequality of banks’ 
market shares might differ substantially between markets yielding the same HHI value. It is possible to 
calculate the numbers equivalent of the HHI as HHIne 1= for every value of the Herfindahl-
Hirschman index (since ( ) ( ) nnnnHHI n
i
111 2
1
2 === å = ), providing evidence that at least two 
different bank size distributions can generate the same HHI. 
 
Hart (1975) takes a slightly different approach to embedding the Herfindahl-Hirschman index into 
distributional theory. Arguing that there are cases where the exact number of banks in an industry is 
 
8 Appendix 3 discusses the Lerner index. 
 8 
 
unknown but information is available about both the banking market’s size and banks’ size 
classification, he proposes to divide the total distribution of bank sizes into classes and to calculate the 
parameters of the original distribution from the parameters of the first moment distribution if the 
relationship between the distributions is known. He obtains a definition of HHI given by: 
 
(2.5) ( ) nHHI 120 += h 
 
where 20h is the coefficient of variation (the possible change of the size structure) of the original 
distribution. As long as the coefficient of variation does not change, an increase in n will result in a 
decrease in the HHI. Hart considers the sensitivity to the entrance of very small banks into an 
oligopolistic market as the greatest disadvantage of the index. 
 
2.2.3 The Hall-Tideman Index and the Rosenbluth Index 
The concentration indices developed by Hall and Tideman (1967) and Rosenbluth (Niehans, 1961) 
resemble one another both in form and in character. Hall and Tideman bring forward a number of 
properties which concentration measures should satisfy and they accept the HHI on the basis of those 
properties. They emphasise the need to include the number of banks in the calculation of a 
concentration index, because it reflects to some extent the conditions of entry into a particular 
industry. 9 Their index takes the form: 
 
(2.6) )12/(1
1å = -=
n
i i
isHTI 
 
where the market share of each bank is weighted by its ranking in order to ensure that the emphasis is 
on the absolute number of banks, and that the largest bank receives weight 1=i . The HTI ranges 
between zero and unity, being close to zero for an infinite number of equal-sized banks, and reaching 
unity in the case of monopoly. Like the Herfindahl-Hirschman index, the HTI equals 1/n for an 
industry with n equally sized banks and the numbers equivalent of the index is defined as HTIne 1= . 
 
Rosenbluth’s index is related in an obvious way to the concentration curve10 but unlike the kCR , it 
takes explicit account of each item in the size distribution. The use of the rankings of banks as weights 
 
9 Entry into an industry is seen as relatively easy, if there are many banks in the industry, and relatively dif-
ficult if the market is shielded by a few banks. This argument will gain emphasis in Section 3.3, which discusses 
the impact of concentration on competition. In empirical investigations of structural models, measures of con-
centration are often used as the only measure approximating market structure. Many studies reveal the impor-
tance of entry barriers for market performance. 
10 The Rosenbluth index and the Gini coefficient are related according to RI=1/(n(1-G)), due to the similarity 
of the Lorenz curve and the concentration curve. See Appendix 3 for explanation of the Gini coefficient and the 
Lorenz curve. Rosenbluth (1955), Kellerer (1960), Niehans (1961) and Marfels (1971b) give details and 
derivations of a theoretical relationship between the concepts of concentration inequality. 
 9 
 
for the calculation of the index, starting with the smallest, makes the index – unlike the kCR – sensitive 
to changes in the bank size distribution of smaller banks. The Rosenbluth index is calculated from the 
area above the concentration curve and below a horizontal line drawn at a level of 100% (entire 
market) on the vertical scale, defined as C in Figures 1 and 2 in Appendix 3. Rosenbluth’s index takes 
the form )2(1 CRI = , which is identical to the HTI for C = 21
1
-å =
n
i i
is . The only difference 
between the indices results from the ranking of banks. For a given degree of inequality, the RI 
decreases with the increase in the numberof banks. Hause (1977) claims that the value of the 
Rosenbluth index is heavily influenced by the small banks in the size distribution, so that its useful-
ness for analysing departures from competition in highly concentrated industries seems quite dubious. 
 
2.2.4 The comprehensive industrial concentration index 
The comprehensive industrial concentration index (CCI) has emerged from the debate on 
concentration and dispersion of banks (or firms) within various industries. The debate rests on two 
arguments. Despite the widely recognised convention that the dominance of the largest few banks 
determines market behaviour, discrete concentration measures have been criticised on the grounds that 
they ignore changes in market structure occurring elsewhere than among the largest banks. 
Conversely, dispersion measures, such as the Lorenz curve and the Gini coefficient, are said to 
undervalue the importance of large banks in the industry. To circumvent the deficiencies of earlier 
indices, Horvarth (1970) presents the CCI, which is capable of reflecting both relative dispersion and 
absolute magnitude. The CCI owes its intellectual heritage to the HHI and takes the form: 
 
(2.7) ))1(1(
2
2
1 i
n
i i
sssCCI -++= å = 
 
It is the sum of the proportional share of the leading bank and the summation of the squares of the 
proportional sizes of each bank, weighted by a multiplier reflecting the proportional size of the rest of 
the industry. The index is unity in the case of monopoly and it is higher than the dominant bank’s 
absolute percentage share for a market with a greater number of banks (Horvarth, 1970). The well-
established theoretical literature on equilibrium subset cartels serve as a formal underpinning for the 
appropriateness of the dissimilar treatment of two classes of banks, i.e. the accentuation of the 
dominant bank at the expense of the rest. 
 
2.2.5 The Hannah and Kay Index 
Hannah and Kay (1977) propose a summary index of concentration of the form: 
 
(2.8) )1/(1
1
)( aa -
=å=
n
i i
sHKI 0>a and 1¹a 
 
 10 
 
where a is an elasticity parameter to be specified and intended to reflect their ideas about changes in 
concentration as a result of the entry or exit of banks, and the sales transfer among the different banks 
in the market. The freedom to choose a allows for alternative views on what is the appropriate 
weighting scheme and for the option to emphasise either the upper or the lower segment of the bank 
size distribution. Therefore, in addition to the distribution of the banks in the market, the value of the 
index is sensitive to the parameter a. For 0®a , the index approaches the number of banks in the 
industry, and for ¥®a , it converges towards the reciprocal of the market share of the largest bank. 
The numbers equivalent of the HKI can be solved for )1/(1))1(( aa -= ee nnHKI as HKIne = . 
 
Hannah and Kay (1977) show that the entry of a bank, whose size is equal to the effective average size 
of the existing banks,11 will result in the greatest reduction in concentration and cause the numbers 
equivalent of the concentration index to increase by one. If a bank of greater than effective average 
size enters the industry, the effect of reduction in concentration would be smaller, or could even be 
reversed (higher concentration because the size effect outweighs the numbers effect). The expansion 
of a bank whose size exceeds the effective average bank size, would increase the value of the HKI, and 
the expansion of a bank smaller than the effective average bank size would reduce it. The distribution 
of the HKI for a given market and for various values of a is presented in Table 1 at the end of this 
section. 
 
2.2.6 The U index 
Davies (1979) defines his U index in terms of inequality and the number of banks in the industry 
reflecting his perception that most of the concentration indices developed earlier attach too much 
weight to either inequality or the number of banks in the market. It takes the form 1-= nIU a , where 
0³a and I is a generally accepted measure of inequality. The index allows flexibility in the weight 
given to size inequality (I) and the number of banks (n) by varying a, which is an innovation compared 
to the concentration measures presented earlier. The choice of a is not arbitrary. To obtain the value of 
a, Davis proposes to estimate a simple model of inter-industry variance in price cost margins of the 
form bap ii C= , where ip and iC are, respectively, the price-cost margin and concentration in 
industry i.12 Replacing iC by the U index and taking logarithms, the regression equation can be 
written as: 
 
(2.9) iiii nI ubbap +++= loglogloglog 21 
 
11 The effective average size is defined as the market size divided by the numbers equivalent calculated from 
the concentration index for a specific industry. 
 11 
 
 
Since bb a=1 and bb -=2 , the estimated value of a can be obtained as 21 bb-=a . 
Theoretically, Davies derives the U index as: 
 
(2.10) ( ) ( ) a
n
i
aa
ii
an
i i
aaan
i i
nsssnnsnU ))(()()(
1
1
1
211
1
2 ååå = -=--= === 
 
by defining inequality as ååå === ===+=
n
i i
n
i i
n
i i
snnXnxxnxcI
1
222
1
22
1
22 /)/(/1 . Hence, I is a 
simple transformation of the coefficient of variation c². Davies investigates the reaction of the U index 
to new entries and mergers in order to shed light on the role and the possible values of the a parameter. 
Since mergers and entry can best be measured by introducing a yardstick measure of typical bank size, 
the Herfindahl average effective size of banks, has been applied.13 Davies proposes that a should be 
constrained to 1£a in empirical applications for the index to deliver economically unambiguous and 
plausible results. For 1£a , a merger always leads to an increase in the recorded concentration and the 
proportionate increase will be in line with the size of the merging banks. The magnitude of the 
parameter a furthermore determines the sensitivity of the index to different types of mergers. U will 
thus become more sensitive to mergers as value of a increases and the merging banks are larger. The 
entry of a new bank into the industry will lead to a fall in recorded concentration, provided the entrant 
is no more than twice the Herfindahl average effective size. The de-concentrating effect is maximised, 
where the entrant is of exactly the average effective size. The entry of very small banks will generally 
have only a minimal effect on concentration, and the magnitude of a determines the sensitivity of U to 
entry. The restriction 1£a , however, implies that ( ) i
aa
i sns <<
-10 . From a theoretical point of 
view, this result is implausible. Oligopolistic behaviour becomes possible only if a is constrained to 
1³a . It therefore appears as if the U index delivers theoretically as well as empirically satisfactory 
results only if 1=a , for which HHIU = . The distribution of the U index for a given market and 
various values of a are presented in Table 1 at the end of this section. 
 
2.2.7 The Hause Indices 
Based on a discussion of the various interpretations of the Cournot model, Hause (1977) introduces six 
criteria that a theoretically reasonable measure of industrial concentration should satisfy. He claims 
that none of the earlier measures meets all of his criteria, and he attempts to derive two measures of 
concentration that do. Both depend on a parameter a that captures the effects of collusion in an 
oligopoly model. Hause furthermore provides numerical evidence that the tendency towards more12 Note that when data on price-cost margins would be available, the obvious question arises why Davies does 
not use these data directly as a measure of competition, rather than to try to fit a model that relates it to some 
other measure. 
13 The Herfindahl effective average size is defined as the industry size divided by the numbers equivalent 
calculated from the Herfindahl-Hirschman index. 
 12 
 
competition as n increases is much slower for low values of a, i.e. a high degree of collusion, than 
implied by the HHI index. The multiplicatively modified Cournot measure takes the form: 
 
(2.11) { }( ) å = --=
n
i
sHHIs
iim
iissH
1
))((2 2,
a
a 
 
where HHI is the Herfindahl-Hirschman index and a is the parameter capturing the degree of 
collusion. To ensure that the index is a decreasing, convex function of the equivalent number of 
equally sized banks implied by the index, it appears necessary to restrict 15.0³a . The a parameter 
decreases inversely to the degree of collusion, and for ¥®a the index approaches the Herfindahl-
Hirschman index. Assuming a two-bank industry, Hause derives the range for the expression raised to 
the power of a as [0, 2(1/3)0.15 = 0.39], so that the exponent on is always exceeds unity (provided a > 
0).14 The validity of the index in theoretical terms depends quite crucially on the specification of a: 
only if ¥®a will the weights attached to the banks’ market shares grow to equal those of the 
Herfindahl-Hirschman index. The index equals one in the monopoly case, it converges to zero for an 
infinite number of equally sized banks in the market and it takes the value ( )
a
)/)1((1 31 nnn -- if the banks 
in the industry are of equal size, converging to the Herfindahl result, i.e. n1 , for large n. Hause 
furthermore proposes the additively adjusted Cournot measure of concentration, which is defined as: 
 
(2.12) { }( ) å = -+=
n
i iiiia
sHHIsssH
1
22 )))(((, bb 
 
with 1>b . This restriction assures the convergence of the index to the Herfindahl-Hirschman index 
for larger numbers of banks, and the measure becomes ))1(( 1211 bb --- -+ nnn , if all banks in the 
market are of equal size. 
 
2.2.8 Entropy Measure 
The Entropy measure has its theoretical foundations in information theory and measures the ex-ante 
expected information content of a distribution. 15 The Entropy measure takes the form:16 
 
(2.13) å =-=
n
i ii
ssE
1 2
log 
 
 
14 Actually, we are not able to reproduce Hause’s upper bound value for the expression raised to the power of 
a. In our view, in a two-bank industry, the expression takes its maximum value, ( )3314 , if 311 =s , which is the 
square of Hause’s maximum. 
15 See Appendix 3 for details. 
16 The old-fashioned base 2 can be replaced by, for instance, the more common base e (for natural 
logarithms), which would change the value of the Entropy by a constant (viz. 2ln1 ). An example for this 
transformation is presented in Appendix 3. 
 13 
 
The index ranges between 0 and n2log , and is therefore not restricted to [0, 1], as most of the other 
measures of concentration presented above. The value of the Entropy varies inversely to the degree of 
concentration. It approaches zero if the underlying market is monopoly and reaches its highest value, 
nE log= , when market shares of all banks are equal and market concentration is lowest. For a given 
number of banks, the index falls with an increase in inequality among those banks (White, 1982), and 
the weights the index attaches to market share decreases in absolute terms as the market share of a 
bank becomes larger (Kwoka, 1985). The numbers equivalent of this index takes the form Een 2= . 
 
2.3 Empirical illustrations of concentration measures 
 
Before concluding the discussion of the various concentration measures, we present a few brief 
numerical exercises taken from practical experience. The first application demonstrates the diverging 
results yielded by the various concentration measures when applied to the same underlying market. 
Table 1 presents the values of these indices based on market shares of 16 banks active in the Dutch 
mortgage market. Even a brief glance reveals the wide spread in these values. The results show clearly 
that not only does the range of possible values dif fer strongly across the indices, but so do the values 
of the indices within this range. For instance, the value is high for the CRk and low for the HHI and in 
particular for the Rosenbluth index, where the ranges of these indices are identical (see Table 1). 
 
Three of the indices, namely the HKI, the multiplic atively modified Hause index and the U index were 
calculated for a number of parameter values for a or a. The sensitivity of these indices to changes in 
those parameter values is displayed in Graphs 1-3. They clearly show that, for a given market 
structure, the U index is an increasing function of the underlying parameters, while the HKI and the 
Hause index are decreasing. The HHI is a special case of the U index for 1=a , whereas the Hause 
index converges to the HHI if ¥®a . 
 
 14 
 
Table 1 Application of concentration measures to the Dutch mortgage market 
Index type Range Parameters Typical features Value 
CR3 Takes only large banks into account; arbitrary cut off 0.82 
CR4 0.90 
CR5 
0 < CRk = 1 
 0.96 
HHI 1/n = HHI = 1 Considers all banks; sensitive to entrance of new banks 0.24 
HTI 0 < HTI = 1 Emphasis on absolute number of banks 0.25 
Rosenbluth 0 < RI = 1 Sensitive to changes in the size distribution of small banks 0.04 
CCI 0 < CCI = 1 Addresses relative dispersion and absolute magnitude; 
suitable for cartel markets 
0.56 
a = 0.005 Sensitive to size distribution; stresses influence small banks 15.79 
a = 0.25 9.41 
a = 5 3.64 
HKI 1/s1 = HKI = n 
a = 10 Sensitive to size distribution; stresses influence large banks 3.40 
a = 0.25 Stresses number of banks 0.09 
a = 1 For a = 1, the U index corresponds to the HHI. 0.24 
a = 2 0.92 
U index 1/n = U = 8 
a = 3 Stresses inequality distribution of banks 3.51 
a = 0.25 Suitable for highly collusive markets 0.44 
a = 1 0.25 
Hause 
index 0 < Hm = 1 
a = 2 Suitable for not collusive markets 0.24 
Entropy 0 = E = log n Based on expected information content of a distribution 2.35 
 
Concentration ratios are often used as input for public policy rules and measures regarding the banking 
market structure. Policy makers can select suitable concentration indices depending on (i) the features 
of their banking market (e.g. the type or level of concentration), (ii) their perceptions regarding the 
relative impact larger and smaller banks have on competition in a certain market, and (iii) their 
perceptions regarding the relative impact of size distribution and number of banks (for instance, 
reflecting the impact of a new entry). These features and perceptions mainly determine which index is 
most appropriate. Concentration ratios are also used to measure the impact of concentration on 
competition. Section 3.2.1 elaborates on the relation between the concentration ratios’ weighting 
schemes and underlying bank conduct. 
 
Graph 1 The Hannah and Kay index (for different values of a) 
0
2
4
6
8
10
12
14
16
0 2 4 6 8 1 0
a
Index value
 
 
 15 
 
Graph 2 The U index (for different values of a) 
0
0,5
1
1,5
2
2,5
3
3,5
4
0 0,5 1 1,5 2 2,5 3
a
Index value
 
 
Graph 3 The multiplicatively-modified Hause index (for different values of a) 
0.2
0.25
0.3
0.35
0.4
0.45
0 0.5 1 1.5 2 2.5 3
aIndex value
 
 
The second practical exercise demonstrates the results for two concentration measures when applied to 
different markets. Table 2 shows the values of the most frequently used indices, the Herfindahl-
Hirschman index and the k bank concentration ratios, for k = 3, 5 and 10, based on market shares in 
terms of total assets of banks in 20 countries (for details and background, see Bikker and Haaf, 2002). 
We have already observed the spread caused by differences in weighting schemes which can occur in 
the (average) levels of the various indices. Differences across countries in the HHI relate most heavily 
to the number of banks (see last column), whereas differences in the CRk are mainly due to the skew-
ness of the bank-size distribution, in particular of its large-bank end. On the whole, apart from a few 
exceptions,17 the rankings of countries are rather similar for the various indices considered, which 
raises confidence in the appropriateness of these indices. The result is typical for what is observed in 
the empirical literature. Surprisingly, the ranking of the HHI and the 3-bank CR bear the closest 
resemblance (with a correlation of 0.98), whereas the ranking of the 5 and 10-bank CRs differ more 
from the HHI (with, respectively, correlations of 0.94 and 0.86). The CR3 and the CR10 are least 
similar. This observation puts into perspective the many ponderous considerations in the literature 
 
17 Exceptions are related mainly to a deviating bank size distribution of the largest 10 banks. This is especially 
true where the total number of banks is limited. 
 16 
 
regarding the neglect of the smaller banks in the k bank CR indices – compared to the HHI, which 
takes all banks into account. 
 
Table 2 Concentration indices for 20 countries, based on total assets (1997) 
 Values Rankings No. of 
Countries HHI CR3 CR5 CR10 HHI CR3 CR5 CR10 banks 
Australia 0.14 0.57 0.77 0.90 6 6 5 6 31 
Austria 0.14 0.53 0.64 0.77 6 9 11 12 78 
Belgium 0.12 0.52 0.75 0.87 9 11 7 7 79 
Canada 0.14 0.54 0.82 0.94 6 8 2 1 44 
Denmark 0.17 0.67 0.80 0.91 4 3 4 5 91 
France 0.05 0.30 0.45 0.64 16 16 16 15 336 
Germany 0.03 0.22 0.31 0.46 18 18 18 19 1,803 
Greece 0.20 0.66 0.82 0.94 3 4 2 1 22 
Ireland 0.17 0.65 0.73 0.84 4 5 8 8 30 
Italy 0.04 0.27 0.40 0.54 17 17 17 17 331 
Japan 0.06 0.39 0.49 0.56 14 14 14 16 140 
Luxembourg 0.03 0.20 0.30 0.49 18 19 19 18 118 
Netherlands 0.23 0.78 0.87 0.93 2 1 1 3 45 
Norway 0.12 0.56 0.67 0.81 9 7 10 11 35 
Portugal 0.09 0.40 0.57 0.82 12 13 12 9 40 
Spain 0.08 0.45 0.56 0.69 13 12 13 13 140 
Sweden 0.12 0.53 0.73 0.92 9 9 8 4 21 
Switzerland 0.26 0.72 0.77 0.82 1 2 5 9 325 
UK 0.06 0.34 0.47 0.68 14 15 15 14 186 
US 0.02 0.15 0.23 0.38 20 20 20 20 717 
Averages 0.11 0.47 0.61 0.75 4,612 
s a 0.07 0.18 0.20 0.18 
Source: Bikker and Haaf (2002); a Standard deviation. 
 
2.4 Concluding remarks 
 
With the exception of the k bank concentration ratio and the Herfindahl-Hirschman index, the 
measures of concentration presented in the preceding discussion have been applied only sparingly in 
the empirical banking literature. The former two indices are also often used as proxies for the market 
structure in structural approaches to measure competition, i.e. the Structure-Conduct-Performance 
paradigm and the efficiency hypothesis, which both will be presented in the next section. The 
Herfindahl-Hirschman index, furthermore, is a statutory measure to evaluate the impact which a 
proposed merger in the US banking industry is to have on the market structure in the region 
concerned. Applied in practice, the various concentration measures may show strongly diverging 
values for the same market, owing to the use of varying weighting schemes, which reflect mainly 
different assessment regarding the relative impact of larger and smaller banks on competition in a 
certain market. Policy makers should choose concentration indices depending on the features of their 
banking market and their perceptions regarding the relative impact larger and smaller banks have on 
competition and regarding the relative impact of size distribution and number of banks. Where for the 
k bank CRs and the HHI the levels may vary, their rankings of banking markets in 20 countries appear 
 17 
 
(as is often the case) to be quite similar. This adds to the suitability of these indices for public policy 
rules and measures. 
 
3 STRUCTURAL MEASURES OF COMPETITION 
 
The literature on the measurement of competition can be divided into two major streams: structural 
and non-structural approaches. The structural approach to the measurement of competition embraces 
the Structure-Conduct-Performance paradigm (SCP) and the efficiency hypothesis, as well as a 
number of formal approaches with roots in Industrial Organisation theory. The two former models 
investigate, respectively, whether a highly concentrated market causes collusive behaviour among the 
larger banks resulting in superior market performance, and whether it is the efficiency of larger banks 
that enhances their performance. These structural models, which link competition to concentration, are 
presented in this section. Non-structural models for the measurement of competition, namely the Iwata 
model (Iwata, 1974), the Bresnahan model,18 and the Panzar-Rosse model (Panzar and Rosse, 1987), 
were developed in reaction to the theoretical and empirical deficiencies of the structural models. These 
New Empirical Industrial Organisation approaches test competition and the use of market power, and 
stress the analysis of banks’ competitive conduct in the absence of structural measures. These non-
structural approaches, which ignore the impact of concentration, will be discussed in Section 3.4. 
 
Structural measures of competition may, in turn, be divided into two major schools of thought: the 
formal and non-formal approaches. The study of the relationship between market performance and 
market structure has its roots in the non-formal framework of the SCP paradigm. Since its origins, this 
framework has evolved largely independently of ongoing refinements in formal models of imperfectly 
competitive markets (Martin, 1993). The empirical application of the SCP paradigm in its original 
form and the recognition that the market structure should be treated as an endogenous variable, led to a 
reformulation of the empirical tests and to attempts to build a formal theoretical framework for the 
structural equations. Oligopoly theory has replaced structure-conduct-performance as the organizing 
framework for industrial economics. In some cases, this work recast structure-conduct-performance 
arguments in a more formal mould (Martin, 1993). Yet, large discrepancies between the formal and 
non-formal approaches remain. 
 
The first part of this section discusses two non-formal approaches to the market structure-market 
performance relationship, the SCP framework and the efficiency hypothesis. These approaches are 
called non-formal as measures for the market structure are not derived theoretically, but chosen at will. 
The second part presents two formal derivations of the competition-concentration relationship, one 
 
18 See Bresnahan (1982) and Lau (1982). This technique is elaborated further in Bresnahan (1989). 
 18 
 
based on the Herfindahl-Hirschman index and one on the k bank concentration ratio. These measures 
of the market structure follow formally from theoretical derivations. Both the formal and the non-
formal approaches link competition to concentration, as in every approach a concentration ratio takesup a central position. 
 
3.1 Non-formal structural approaches to competition 
 
The SCP and the efficiency hypothesis are the two most common non-formal structural approaches to 
measure the impact of concentration on competition. In its original form, the SCP explains market 
performance as the result of an exogenously given market structure, which depends upon basic 
demand and supply side conditions (Reid, 1987, and Scherer and Ross, 1990) and which influences the 
conduct of banks in the industry. A higher level of concentration in the market is assumed to foster 
collusion among the active banks and to reduce the degree of competition in that particular market. 
The application of the SCP to the banking literature has been criticised by various authors, for instance 
by Gilbert (1984), Reid (1987), Vesala (1995) and Bos (2002). Their criticism is directed at the form 
of the model rather than at the specification of the variables used. Much of the criticism is related to 
the one-way causality – from market structure to market performance – inherent in the original model 
as it is still being applied in many banking studies, and to the failure by recent studies to incorporate 
new developments in the theory of industrial organisations. The extensive literature applying the 
structure-performance paradigm to the banking industry has been summarised by, for instance, Gilbert 
(1984) and Molyneux et al. (1996). In this context, it should be noted that most of the studies applying 
the SCP framework to the banking industry do not take explicit account of the conduct of banks.19 
This being the case, the remaining of the analysis will focus on the structure-performance (S-P) 
relationship. 
 
The efficiency hypothesis, developed by Demsetz (1973) and Peltzman (1977), challenges the line of 
reasoning of the traditional S-P paradigm and offers a competing explanation of the relation between 
market structure and performance. The hypothesis claims that if a bank achieves a higher degree of 
efficiency than other banks in the market (i.e. its cost structure is comparatively more effective), its 
profit maximising behaviour will allow it to gain market share by reducing prices (Molyneux and 
Forbes, 1995). Market structure is therefore shaped endogenously by banks’ performance, so that 
concentration is a result of the superior efficiency of the leading banks (Vesala, 1995). 
 
 
19 To the best of our knowledge, only the study by Calem and Carlino (1991) takes explicit account of bank 
conduct in the empirical application of the SCP framework. They found that strategic conduct of commercial and 
federal savings banks in the United States was not limited to concentrated markets. Chapter 4, Section 4.4, 
investigates the structure-conduct relationship. 
 19 
 
Empirically, one may distinguish the S-P paradigm from the efficiency hypothesis by looking at the 
endogenous variable measuring the performance of a particular bank, which is usually estimated as a 
function of exogenous market structure and control variables, as in: 
 
(3.1) k tjik ktitjti XMSCR ,,2,2,10, å ++++=P aaaa 
 
ti ,P is a performance measure for bank i, tjCR , is a measure of concentration in region j (the region 
to which bank i belongs) and tiMS , is the market share of bank i; t refers to period t. CR and MS each 
proxies an aspect of the market structure. kX is a vector of control variables included to account for 
both bank-specific and region-specific characteristics (Molyneux and Forbes, 1995). The traditional 
structure-performance relationship would apply to the data if 01 >a and 02 =a , and the efficiency 
hypothesis holds if 01 =a and 02 >a . 
 
Resulting from the definition of bank performance measures in various empirical studies, the S-P 
literature as applied to the banking industry can be divided into two groups (Gilbert, 1984, Molyneux 
et al., 1996, and Bos, 2002). A great variety of articles have employed the price of a particular product 
or service as a measure of bank performance, whereas a smaller number of studies relied on 
profitability measures, such as banks’ profit rates. 
 
The first group uses a price measure of some kind – commonly either average prices of products or 
services (average interest rates on loans or deposits, average service charges on demand deposits), or 
the price of particular products or services (e.g. business loans). The use of average prices as a proxy 
for bank performance has had its share of criticism. More precisely defined product markets are 
examined when data on individual product prices is employed. Price measures, however, ignore the 
(possible) cross-subsidisation that is not uncommon for a multi-product bank, so that studies applying 
these performance measures might present a misleading picture as compared to the overall 
performance of the bank (Molyneux and Forbes, 1995). The definition of relevant product (and 
geographical) markets, as discussed in the introduction, is also an issue here. 
 
Apart from the ease with which they can be obtained, profitability measures have the advantage of 
consolidating product profits and losses of a multi-product bank into one single figure; i.e. they do 
include cross-subsidisation among products. Like the pricing approach to banking performance, the 
calculation of profit measures from both stock and flow variables is a drawback (Molyneux et al., 
1996). In addition, Vesala (1995) claims that banks’ operational inefficiencies can lead to lower 
profitability, so that market power and profits need not necessarily be positively correlated. 
 20 
 
Furthermore, the calculation of an aggregated profit measure precludes the estimation of market power 
in individual markets. 
 
Table 3 summarises a variety of measures that can be used to describe market structure in the banking 
industry. In most studies, banking market structure is proxied by a measure of concentration and the 
market shares of the banks in the market.20 The studies applying the S-P to the US banking industry 
usually define market structure in terms of the Herfindahl-Hirschman index, the k bank concentration 
ratio or the number of banks in the market (Molyneux et al., 1996). Only a very limited number of 
studies apply the Entropy, the Hall-Tideman index or the Gini coefficient as market structure measure. 
Not all structure-performance studies account for barriers to entry. 
 
Table 3 The Structure -Performance relationship in banking 
Concentration 
- Herfindahl-Hirschman index 
- k bank concentration ratio 
- number of banks 
- other concentration ratios or Gini coefficient 
Market sharea - defined for the relevant market 
Barriers to entryb 
- regulatory barriers: bank charters or branches and the 
conditions related to them 
- non-regulatory barriers: minimal efficient size, product 
differentia tion, scale economies, technology and know-
how 
Market struc-
ture 
(exogenous) 
Number of branches - operating in the relevant market 
 
Price measures 
- price of particular products or services (for instance 
business loans) 
- average prices of products or services (e.g. average interest 
rates on loans; average interest rates on deposits; average 
service charges on demand deposits) 
Performance 
(endogenous) 
Profitability measures - single figure consolidating profits and losses realised with 
all products offered by a bank 
a Market shares of individual banks are very often used as a measure indicating the efficiency of banks; b Vesala (1995) 
claims that the correlation between regulatory protection and concentration, as well as the correlation between regulatoryprotection and profitability may explain the detected positive correlation between concentration and profitability. 
 
The control variables usually included in the empirical analysis proxy market demand conditions (e.g. 
per capita income or wage, levels of population density, immigration into specific markets), cost 
differences across banks, size-induced differences between banks, such as scale economies (e.g. 
measure of individual bank size), different risk categories (Molyneux et al., 1996) and ownership 
differences (Lloyd-Williams et al., 1994). 
 
The results of the rather limited number of studies investigating the structure-performance relationship 
for European banking markets are no less ambiguous than those obtained from the US banking 
 
20 Market share is often also considered as a proxy for bank efficiency. 
 21 
 
industry (Gilbert, 1984, Weiss, 1989, Molyneux et al., 1996, and Bos, 2002). The scarcity of those 
studies is mainly the result of the lack of sub-market banking data for the European banking markets, 
which makes it extremely difficult to define a meaningful (relevant) market area and a reasonable 
measure of concentration in universal banking and nation-wide banking markets. 
 
3.2 Formal structural approaches to competition 
 
The growing body of literature subjecting the choice of profitability measures to formal analysis has 
been summarised by Martin (1993). Most of these formal studies generalise the Lerner index of 
monopoly power. They do not stress any one measure of profitability as correct or the best, but 
provide guidance in selecting tests of market power and profitability. 
 
The overall organising framework, which will now be introduced, has its roots in Industrial 
Organisations theory. It provides the basis for the discussion of formal structural and non-structural 
models in competition theory. The derivations are based on the profit maximisation problem for 
oligopolistic markets presented, for instance, by Cowling (1976) and Cowling and Waterson (1976). 
There are n unequally sized banks in the industry producing a homogeneous product.21 Bank size 
differences are incorporated by the shape of the individual banks’ cost functions. The profit function 
for an individual bank takes the form: 
 
(3.2) iiiii Fxcpx --=P )( 
 
where ? i is profit, xi is the volume of output, p is the output price, ci are the variable costs and Fi are 
fixed costs of bank i. Banks are assumed to face downward sloping market demand functions, the 
inverse of which is defined as: 
 
(3.3) )...()( 21 nxxxfXfp +++== 
 
The first order condition for profit maximising of bank i yields: 
 
(3.4) 0)()/)((/ =¢-¢+=P iiiiii xcxdxdXXfpdxd 
 
and can be written as: 
 
(3.5) 0)()1)(( =¢-+¢+ iii xcxXfp l 
 
 
21 For a discussion on homogeneity, see Stigler (1964). 
 22 
 
with i
n
ij ji
dxxd /å ¹=l . This variable il is called the conjectural variation of bank i and is the 
change in output of all remaining banks anticipated by bank i in response to an initial change in its 
own output idx . The concept of conjectural variation can parameterize any static or dynamic 
equilibrium structure in which firms’ reaction functions are continous. It allows differentiation 
between various market forms, and can be viewed as including expectations about the behaviour of 
potential entrants as well as actual rivals (Cowling, 1976, p.5). Depending on the underlying market 
form, il can take values between –1 and i
n
ij j
xx /å ¹ . In the case of perfect competition, an increase 
in output by one bank has no effect on the market price and quantity: ( )iidxdX l+== 10 , and hence 
1-=il . A bank operating in a Cournot oligopoly does not expect retaliation from its competitors, i.e. 
it expects other banks to remain inactive in response to an increase in its own output. An increase in 
output by bank i thus leads to an increase in total industry output by the same amount: 
( )iidxdX l+== 11 , so that 0=il . In the case of perfect collusion, a bank i expects full retaliation 
from its competitors, who intend to protect their market share in response to an increase in output by 
bank i: == ii xXdxdX il+1 (an increase in output by bank i by one unit leads to an increase in 
market output by ixX units), and =il ( ) i
n
ij jii
xxxxX /å ¹=- . In what follows, equation (3.5) 
will serve as a starting point both for the discussions of a variety of measures of competition and for 
the formal derivations of the relationship between measures of concentration and measures of 
competit ion. 
 
3.2.1 The Herfindahl-Hirschman index in formal structure-performance models 
Multiplying equation (3.5) with xi and summing the result over all banks yields: 
 
(3.6) ( ) ( ) 0)/()/(
1
222
11
=¢-¢+ ååå ===
n
i iiiii
n
i
n
i i
xxcXxXdxdXXfpx 
 
and rearranging gives the price-cost margin as measure of performance for the industry as: 
 
(3.7) ( ) )/)(/()/(/))(( 2
1
2
1 i
n
i iiii
n
i i
dxdXpXXXfXxpXxxcpx ¢-=¢- åå == 
 
which can be rewritten as: 
 
(3.8) D
n
i i
n
ij jiii
n
i iiii
dxxddxdxspXxxcpx h/)())((
1
2
1 å åå = ¹= +-=¢- 
 
 23 
 
or, as ( )XXfpdpXdXpD ¢==h ,22 
 
(3.9) ( ) D
n
i iiii
HHIpXxxcpx hg /1/))((
1
+-=¢-å = 
 
where åå ===
n
i i
n
i ii
xx
1
2
1
2 /lg . This expression represents the average price-cost margin in terms of 
Dh , the price elasticity of demand, the Herfindahl-Hirschman index HHI and g, a term capturing the 
conjectural variation. This derivation conforms with the assumptions of the SCP framework in that a 
higher degree of concentration in an industry results in higher price-cost margins. In addition, this 
theoretical derivation justifies the use of the Herfindahl-Hirschman index as a measure of 
concentration in structure-performance relationships, when g is known and equal for all banks. 
 
In line with the general derivations presented above, Dickson (1981) introduced the conjectural 
variation elasticity ( ) ( )iii xdxXdXt = , i.e. the proportional change in market output anticipated 
by bank i ( )i" . On the basis of this definition, two statements are possible. First the degree of 
expected retaliation allows to draw conclusions about the underlying market structure. Second, there is 
theoretical justification for concentration measures to determine price-cost margins. Dickson redefined 
the price-cost margin of equation (3.9), ignoring the constant Dh , as: 
 
(3.10) å ==
n
i ii
tsA
1
 
 
with ( )iii st l+= 1 . Note that: ( ) +=+=+= ååå === HHIXxHHIsts
n
i ii
n
i ii
n
i ii 1
22
1
2
1
/1 ll 
( )gg +=å = 1/ 21 2 HHIXx
n
i i
. Just like the definition of the conjectural variation il , the conjectural 
variation elasticity it takes different values depending on the underlying market structure. First, in a 
Cournot oligopoly, XdxXdX i= , such that )/)(/()( XxXdxXdXt iii = = si. Second, 1=it in 
the case of perfect collusion, since ii xdxXdX = . Third, for perfectly competitive markets 
0=idxdX and 0=it . Measures of concentration can therefore be treated as weighted averages of 
conjectural variation elasticities. 
 
The results obtained by Dickson for theoretically justifiable measures of concentration presented in 
Section 3.2 are summarised in Table 4. It turns out that the conjectural variation of a bank can take a 
fixed value, or vary within a certain range. The value of the conjectural variation elasticity thus tells us22 We assume Dh to have a negative value. 
 24 
 
how the market structure has interacted with firms’ decisions to give us a measure that is more directly 
related to the objectives of policy makers (such as welfare). 
 
Table 4 Concentration measures and their conjectural variations 
Measure of 
concentration 
Weight wi Range of conjectural variation elasticity ti
 a) 
HHI wi = si wi = t i = si 
wi = 1 for i £ k wi = t i = 1 CRk wi = 0 for i > k wi = t i = 0 
wi = 1 for s1 and wi = t i = 1 CCI 
wi = (si + (si - si
2)) for i > 1 si < w i (= t i ) £ 0.75 
HKI b) wi = si
 a – 1 0 £ ti £ 1 
 c) 
wi = si n
(a – 1) / a 0 < si n
(a –1) / a (= ti ) < si for a £ 1 and U 
 ti > 1 is possible for a > 1 
Hm wi = s
a))((1 2ii sHs
i
-- si < t i < 1 
a) See the text for the market structure corresponding to each of the values of ti; b) For the derivation of this weight, Dickson 
(1981) departs from the expression HKI = S sia from which Hannah and Kay derived their index as presented in equation 
(2.8); c) The index can be made compatible by restricting 0 £ sia - 1 £ 1, and a ³ 1. The parameter a is equal to 1 in the monop-
oly case, and the Cournot equilibrium is obtained when a ® 2. 
 
3.2.2 The k bank concentration ratio in formal structure-performance models 
Another approach to the theoretical derivation of the SCP relationship has been proposed by, among 
others, Saving and Geroski (Reid, 1987) and assumes a k bank cartel with fringe competitors. Of the n 
banks in a market, k banks are acting as a cartel, and the remaining n-k banks are the competitive 
fringe in the industry. These n-k banks are price takers and equilibrate price and marginal cost, 
icp = , nki ),...,1( +=" , for profit maximising purposes. The supply of these banks can then be 
written as ( )pc i 1- , so that the aggregate supply of the competitive fringe is ( ) å += -- =
n
ki ikn
pcpS
1
1 )( . 
Industry demand is defined by ( )pDT , with ( ) 0<¢ pDT . Under the conditions defined above, the k 
bank cartel faces demand ( ) ( ) ( )pSpDpD knTk --= , where ( ) 0>¢- pS kn and ( ) 0<¢ pDT . Differen-
tiating the above equation and dividing by ( ) ppDk yields ( ) ( ) ( ) ( )-¢=¢ pDppDpDppD rTkk // 
( ) ( )pDppS rkn /-¢ which can be written as ( ) ( ) ( ) ( )pDpSpDpD rknSrTDD knTk // ---= hhh , where 
kD
h , 
TD
h and 
knS -
h are, respectively, the elasticities with respect to price of residual demand, industry 
demand and fringe supply. For profit maximising reasons, the k bank cartel should, in line with the 
concept of the Lerner index, set its price-marginal cost margin equal to the reciprocal of the elasticity 
of its demand curve: 
 
(3.11) ( ) ( ) ( )kknSTD
k
kknknSkTTDkD
j
C
C
DSDDp
cp
--
=
-
==
¢-
---
1
11
hhhhh
 
 
 25 
 
since kkT CDD /1/ = and kkkkkTkkn CCCDDDDS /)1(1/1/)(/ -=-=-=- . This result 
shows that the application of the k bank concentration ratio for empirical estimations of the structure-
performance relationship is theoretically justified for a market with a k bank cartel and n-k small 
banks. Hence, if we assume that the market is dominated by a cartel, in a SP relationship, we should 
use CRk. 
 
3.3 Concluding remarks 
 
This section discussed formal and non-formal approaches to the relationships between market 
structure and market performance. The non-formal structure-performance paradigm and efficiency 
hypothesis, although lacking formal back-up in micro-economic theory, have frequently been applied 
to the banking industry and provide policy makers by measures of market structure and performance 
as well as their interrelationship. The formal approaches presented give evidence of the 
appropriateness of the CRk and the HHI in the empirical application of structure-performance models. 
These formal relationships also state that bank conduct plays a role: CRk is based on the assumption of 
a k bank cartel, HHI presupposes that conduct can be described by conjectural variation and similar 
deductions hold for other concentration ratios. These interpretations help policy makers to select a 
measure of concentration appropriate to their needs. 
 
4 NON-STRUCTURAL MEASURES OF COMPETITION 
 
This section discusses three non-structural measures of competition, namely the Iwata model, the 
Bresnahan model and the Panzar and Rosse approach. The derivations of the first two measures are 
based on the results obtained for the oligopoly profit-maximisation problem presented in Section 3.3. 
The Panzar and Rosse method is based on the comparative static propertie s of the reduced-form 
revenue approach. 
 
4.1 The Iwata Model 
 
The Iwata model allows the estimation of conjectural variation values for individual banks supplying a 
homogeneous product in an oligopolistic market (Iwata, 1974). Although, to the best of our 
knowledge, this measure has been applied to the banking industry only once,23 it is included in the 
present overview for completeness’ sake. Defining the price elasticity of demand as 
=Dh )/)(/( pxdXdp- , equation (3.5) can be written as ( ) ( ) 01)/)(/1( =¢-+- iiiD xcxxpp lh , 
which, by re-arranging yields: 
 
23 Shaffer and DiSalvo (1994) apply this model to a two-banks market. 
 26 
 
 
(4.1) ( ) 1)/)(/)(( --¢= iiDi xXppxchl 
The numerical value of this conjectural variation will be obtained indirectly. Under the assumptions 
that p and Xxi are strict functions of exogenous variables, and that Dh , the elasticity of demand, is 
constant, the method involves the estimation of a market demand function and cost functions of 
individual banks to obtain a numerical value of the conjectural variation for each bank. The 
application of this model to the European banking industry is difficult, especially given the lack of 
micro-data for the structure of cost and production for homogeneous products of a large number of 
players in the European banking markets. 
 
4.2 The Bresnahan model 
 
Bresnahan (1982) and Lau (1982) present a short-run model for the empirical determination of the 
market power of an average bank. Based on time-series of industry data, the conjectural variation 
parameter =l ndxxd iji j /)/1( å ¹+ , with 10 ££ l , is determined by simultaneous estimations of 
the market demand and supply curves. Banks maximise their profits by equating margina l cost and 
perceived marginal revenue. The perceived marginal revenue coincides with the demand price in 
competitive equilibrium and with the industry’s marginal revenue in the collusive extreme (Shaffer, 
1993). Under the bank equality assumption of the Bresnahan model, equation (3.5) can be rearranged 
to yield ( ) ( )ixcXXfp ¢=¢+ l . Consistent with the statements about market forms and the 
corresponding conjectural variations in the introduction to this section, 0=l for the average bank in 
a perfectly competitive market. Under Cournot equilibrium, 0=å ¹ iji j dxxd , so that n1=l and 
( ) =×+ nhp / ( )ixc¢ , with ( ) ( )XXfh ¢=× , the semi-elasticity of market demand. ( ) ( )ixchp ¢=×+ for 
perfect collusion since =+= å ¹ ndxxd iji j /)1(l 1)/(/)/)(1( ==-+ iii nxxnxxX , i" . 
 
Empirical applications of the Bresnahan model are rather scarce. It has been estimated by Shaffer 
(1989 and 1993) for, respectively, the US loan markets and for the Canadian banking industry. 
Suominen (1994) applied the model in its original one-product version to the Finnish loan market for 
the period 1960-84 during which the interest rates applied by banks were tightly regulated. Interest 
rates on loans were deregulated in August 1986, but interest rateson deposits remained effectively 
restricted until 1990. An adapted two-product version is applied to the Finnish loan market for the 
period after deregulation (September 1986-December 1989). 
 
 27 
 
Suominen finds coefficient estimates for ? which are close to zero and not significantly different from 
zero at the 5 per cent level for the period with regulated interest rates in both markets, and values of ? 
indicating use of market power after the deregulation of the loan market. Swank (1995) estimated 
Bresnahan’s model to obtain the degree of competition in the Dutch loan and deposit markets over the 
period 1957-90, and found that both markets were significantly more oligopolistic than in Cournot 
equilibrium. 
 
Chapter 5 presents applications of the Bresnahan model to loans markets as well as deposits markets in 
nine European countries over the latest two or three decades. Where values of ? appear to be 
significant different from zero, so that perfect competition should be rejected, they are nevertheless 
close to zero. In many cases, the hypothesis ? = 0 (i.e. perfect competition) can not be rejected. 
 
4.3 The Panzar and Rosse approach 
 
The method developed by Panzar and Rosse (1987) determines the competitive behaviour of banks on 
the basis of the comparative static properties of reduced-form revenue equations based on cross-
section data. Panzar and Rosse (P -R) show that if their method is to yield plausible results, banks need 
to have operated in a long-term equilibr ium (i.e. the number of banks needs to be endogenous to the 
model) while the performance of banks needs to be influenced by the actions of other market 
participants. Furthermore, the model assumes a price elasticity of demand, e, greater than unity, and a 
homogeneous cost structure. To obtain the equilibrium output and the equilibrium number of banks, 
profits are maximised at the bank as well as the industry level. That means, first, that bank i maximises 
it profits where marginal revenue equals marginal cost: 
 
(4.2) ( ) ( ) 0,,,, =¢-¢ iiiiiii twxCznxR 
 
ix being the output of bank i, n the number of banks, iw a vector of m factor input prices of bank i, 
iz a vector of exogenous variables that shift the bank’s revenue function, and it a vector of 
exogenous variables that shift the bank’s cost function. Secondly, it means that, in equilibrium, the 
zero profit constraint holds at the market level: 
 
(4.3) ( ) ( ) 0,,,, ***** =- twxCznxR ii 
 
Variables marked with * represent equilibrium values. Market power is measured by the extent to 
which a change in factor input prices (
ik
w¶ ) is reflected in the equilibrium revenues ( *iR¶ ) earned by 
 28 
 
bank i. Panzar and Rosse define a measure of competition, the ‘H statistic’ as the sum of the 
elasticities of the reduced form revenues with respect to factor prices:24 
 
(4.4) )/)(/( *
1
*
ikk
m
k i
RwwRH
ii
¶¶= å = 
 
The estimated value of the H statistic ranges between 1£<¥- H . H is smaller than zero if the 
underlying market is monopoly, it ranges between zero and unity for monopolistic competition, and an 
H of unity indicates perfect competition. Shaffer (1983) demonstrated formal linkages between the 
Panzar-Rosse H statistic, the conjectural variation elastic ity and the Lerner index. Only a limited 
number of studies tests the P-R method for the banking industry. Table 5 summarises the results of 
those investigations. 
 
Table 5 P-R model results in other studies 
Authors Period Countries considered Results 
Shaffer (1982) 1979 New York monopolistic competition 
Nathan and Neave (1989) 1982-84 Canada 1982: perfect comp.; 1983-
84: monopolistic comp. 
Lloyd-Williams et al. (1991) 1986-88 Japan monopoly 
Molyneux et al. (1994) 1986-89 France, Germany, Italy, Spain 
and UK 
mon.: Italy; mon. comp.: 
France, Germany, Spain, UK 
Vesala (1995) 1985-92 Finland monopolistic competition for 
all but two years 
Molyneux et al. (1996) 1986-88 Japan monopoly 
Coccorese (1998) 1988-96 Italy monopolistic competition 
Rime (1999) 1987-94 Switzerland monopolistic competition 
Bikker and Groeneveld (2000) 1989-96 15 EU countries monopolistic competition 
De Bandt and Davis (2000) 1992-96 France, Germany and Italy 
large banks: mon. comp. in 
all countries; small banks: 
mon. comp. in Italy, monop-
oly in France, Germany 
Chapter 5 1988-98 23 OECD countries monopolistic competition 
 
Shaffer (1982), in his pioneering study on New York banks, observed monopolistic competition. For 
Canadian banks, Nathan and Neave (1989) found perfect competition for 1982 and monopolistic 
competition for 1983-84. Lloyd-Williams et al. (1991) and Molyneux et al. (1996) revealed perfect 
collusion for Japan. Molyneux et al. (1994) tested the P-R statistic on a sample of French, German, 
Italian, Spanish and British banks for the period 1986-89 in order to assess the competitive conditions 
in major EC banking markets. They obtain values for H which are not significantly different from zero 
and from unity for France, Germany (except for 1987), Spain and the UK, thus pointing to 
monopolistic competition. The H-statistic for Italy during 1987-89 is negative and significantly 
different from zero, hence it was not possible to reject the hypotheses of monopoly. Coccorese (1998), 
however, who also intends to evaluate the degree of competition in the Italian banking sector, obtains 
 
24 See Panzar and Rosse (1987) or Vesala (1995) for details of the formal derivation of the H statistic. 
 29 
 
significantly non-negative values for H. H was also significantly different from unity, except in 1992 
and 1994. 
 
Vesala (1995) applies the model to the Finnish banking industry (1985-92) to test for competition and 
market power in the Finnish banking sector. His estimates of H were always positive, but significantly 
different from zero and from unity only in 1989 and 1990. For Switzerland, Rime (1999) observed 
monopolistic competition. Bikker and Groeneveld (2000) determine the competitive structure of the 
whole EU banking industry. The estimated values for the H-statistic lie between two thirds and one in 
most countries. The hypothesis 0=H is rejected for all countries, whereas 1=H cannot be rejected 
for Belgium and Greece at the 95% confidence level. De Brandt and Davis (2000) investigate banking 
markets in France, Germany and Italy within groups of large and small banks. Aiming to assess the 
effects of EMU on market conditions, they obtain estimates of H, which are significantly different 
from zero and from unity for large banks in all three countries. The H statistics estimated for the 
sample with small banks indicate monopolistic competition in Italy, and monopoly power in France 
and Germany. 
 
Chapter 4 considers banks in 23 OECD countries and investigates small, medium-sized and large 
banks separately. This P-R analysis finds monopolistic competition virtually everywhere, although 
perfect competition can not be rejected for some market segments. 
 
5 CONCLUSIONS 
 
This chapter surveys the various approaches to the measurement of concentration and competition, 
which are fundamental for welfare-related public policy toward structure and conduct in banking 
markets. The presentation and analysis of ten measures of concentration was supplemented by 
numerical examples illustrating their properties when applied empirically. The choice of the 
concentration index is mainly dependent on the policy makers’ perception of the relative influence on 
competition attached to large and small banks. The Herfindahl-Hirschman index and the k bank